F Uniform Streaming Flow past a Solid Sphere - Stokes Law

UNIFORM STREAMING FLOW PAST A SOLID SPHERE - STOKES LAW 

The analysis of the preceding section was carried out by use of a spherical coordinate system, but the majority of the results are valid for an axisymmetric body of arbitrary shape. The necessity to specify a particular particle geometry occurs only when we apply boundary conditions on the particle surface (that is, when we evaluate the coefficients Cn and Dn in the spherical coordinate form of the solution). For this purpose, an exact solution requires that the body surface be a coordinate surface in the coordinate system that is used, and this effectively restricts the application of (7-149) to streaming flow past spherical bodies, which may be solid, as subsequently considered, or spherical bubbles or drops, as considered in section H. 

we consider Stokes problem of uniform, streaming motion in the positive z direction, past a stationary solid sphere. The problem corresponds to the schematic representation shown in Fig. 7-11 when the body is spherical. This problem may also be viewed as that of a solid spherical particle that is translating in the negative z direction through an unbounded stationary fluid under the action of some external force. From a frame of reference whose origin is fixed at the center of the sphere, the latter problem is clearly identical with the problem pictured in Fig. 7-11. Because we have already derived the form for the stream-function under the assumption of a uniform flow at infinity, we adopt the latter frame of reference. The problem then reduces to applying boundary conditions at the surface of the sphere to determine the constants C and Dn in the general equation (7-149). The boundary conditions on the surface of a solid sphere are the kinematic condition and the no-slip condition, 

In terms of the streamfhnction, these conditions require that 

It follows from the first of these conditions that // = const on the sphere surface, and, because f = 0 at i] = 1 for all r (because of symmetry), we can see that the appropriate version of the first condition in (7 153) is 

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F. Uniform Streaming Flow past a Solid Sphere - Stokes Law... 

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